This type of medical suture-grasping instrument is employed in endoscopic surgery in order to allow the surgeon, by grasping the suture material, to produce sutures inside the body. A generic medical instrument is disclosed, for instance, in patent DE 198 48 958 A1. With this known suture-grasping instrument, the rigid jaw member of the tool comprises a recess that runs perpendicularly through the jaw member and that serves for inserting the suture material that is to be grasped with the tool closed. With this known instrument it is possible to catch the suture material well, but the recess configured in the rigid jaw member is too large to be able to guide the suture material precisely to the suturing position that is to be configured.